Grad Auction Theory

Auction Theory: Four Canonical Formats and Revenue Equivalence

Overview

Auction theory analyzes strategic bidding behavior across different selling mechanisms. The four canonical formats — English (ascending), Dutch (descending), first-price sealed-bid, and second-price sealed-bid (Vickrey) — generate identical expected revenue under standard assumptions. The Revenue Equivalence Theorem (RET) is the central benchmark; deviations from its assumptions drive all practical auction design decisions.

When to Use

• Choosing among auction formats for selling goods, spectrum, procurement, or ad slots
• Analyzing bidder strategy (bid shading, sniping, jump bidding) under a specific format
• Evaluating whether a proposed auction achieves optimal revenue or efficiency

When NOT to Use

• Posted-price or negotiated sales where no competitive bidding occurs
• Multi-unit auctions with complex complementarities (use combinatorial auction frameworks)
• The seller has no commitment power to enforce auction rules

Assumptions

IRON LAW: Revenue equivalence holds ONLY with risk-neutral bidders,
independent private values (IPV), and symmetric bidders — violate ANY
assumption and auction format matters.

• Single indivisible object for sale
• Bidders are risk-neutral expected-utility maximizers
• Values are independently and identically distributed (IPV)
• Bidders are symmetric (same distribution of values)
• Payment is a function of bids only (no externalities)

Methodology

Step 1 — Identify the Value Model

Classify the setting: independent private values (IPV), common values, or affiliated values. IPV means each bidder knows their own value with certainty; common values mean the object has one true value unknown to all; affiliated values generalize both.

Step 2 — Derive Equilibrium Bidding Strategies

For each auction format, solve for the Bayesian Nash equilibrium. In second-price / English auctions under IPV, bidding true value is dominant. In first-price / Dutch auctions, bidders shade below true value: b(v) = E[Y1 | Y1 < v] where Y1 is the highest competing value.

Step 3 — Apply Revenue Equivalence or Identify Violations

Under RET assumptions, all four formats yield the same expected revenue. Check for violations: (a) risk aversion favors first-price over second-price; (b) asymmetric bidders break equivalence; (c) common values introduce the winner's curse and favor English auctions (Milgrom-Weber linkage principle); (d) budget constraints, entry costs, or reserve prices create format-dependent effects.

Step 4 — Recommend Format and Parameters

Given the identified deviations, recommend the auction format, optimal reserve price (r* where r* = v0 + [1 - F(r*)] / f(r*) in the IPV case), and any additional design features (e.g., entry fees, information disclosure policy).

Output Format

## Auction Analysis: [Context]

### Value Model
- **Type**: IPV / Common / Affiliated
- **Distribution**: [bidder value distribution]
- **Number of bidders**: [N]
- **Risk attitude**: risk-neutral / risk-averse

### Format Comparison
| Format              | Equilibrium Strategy         | Expected Revenue | Notes        |
|--------------------|------------------------------|-----------------|--------------|
| English (ascending) |                              |                 |              |
| Dutch (descending)  |                              |                 |              |
| First-price sealed  |                              |                 |              |
| Second-price sealed |                              |                 |              |

### Revenue Equivalence Assessment
- **RET holds?** Yes / No
- **Violation source**: [risk aversion / asymmetry / common values / other]
- **Ranking**: [which format generates highest revenue and why]

### Optimal Reserve Price
- **r*** = [value]
- **Derivation**: [brief]

### Recommendation
[Chosen format, reserve price, and rationale]

Gotchas

• The winner's curse is a common-value phenomenon — it does not apply in pure IPV settings, yet bidders often behave as if it does
• Revenue equivalence is about *expected* revenue; variance differs across formats (first-price has lower variance)
• Optimal reserve price excludes some efficient trades — the seller sacrifices efficiency for revenue
• In practice, English auctions reveal more information, which helps with affiliated values (Milgrom-Weber linkage principle)
• Collusion is easier in second-price and English auctions; first-price is more robust to bidder rings
• Online auctions (eBay-style) are not pure English auctions — hard close times, proxy bidding, and sniping change equilibrium behavior

References

• Vickrey, W. (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders." *Journal of Finance*.
• Riley, J. & Samuelson, W. (1981). "Optimal Auctions." *American Economic Review*.
• Milgrom, P. & Weber, R. (1982). "A Theory of Auctions and Competitive Bidding." *Econometrica*.
• Krishna, V. (2010). *Auction Theory*, 2nd ed. Academic Press.